1236 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



the dielectric loading; this assumption obviates the somewhat laborious 

 calculation of j8i for each spurious mode from (40) . Thus, 



/3o - /3i ^ /?(, - hi , (96) 



and a good estimate of the maximum power which crosstalks into any- 

 spurious mode is 



(Pl)max ^ [2(C + d)/{ho - h)]\ (97) 



If the form and dimensions of a bend compensator are fixed, and the 

 TEoi — TMii decoupling condition is assumed to be met by adjusting the 

 permittivity, it turns out that the maximum power which crosstalks 

 into a given mode is proportional to (a/b)'. It is thus easy to calculate 

 the bending radius which makes (Pi) max for any given mode equal to a 

 (small) preassigned value. The total power abstracted from the TEoi 

 mode by mode conversion will be a complicated, fluctuating function of 

 distance along the bend, or of frequency at a fixed distance, because of 

 the different critical distances for maximum power transfer into the dif- 

 ferent spurious modes, but we can get an idea of the minimum tolerable 

 bending radius by considering just the crosstalk into the one or two most 

 troublesome modes. 



It seems likely that with present-day dielectrics at milhmeter Avave- 

 lengths, dielectric losses in a compensated bend will be comparable to i 

 mode conversion losses. For this reason an estimate of dielectric losses 

 is given in connection with each type of compensator design discussed 

 below. 



2.3.1 The Geometrical Optics Solution 



An obvious way to design a bend compensator on paper is to load the 

 bend with a medium of continuously varying permittiA'ity for which* 



8 = -2(p/fo) cos^. (98) 



This may be called the geometrical optics solution of the bend problem, 

 since to first order it equalizes the optical length of all circular paths 

 which are coaxial mth the curved center line of the waveguide. Physi- 

 cally the required variation of permittivity is rather simple; the per- 

 mittivity at each point depends only on the distance of the point from a 

 line through the center of curvature perpendicular to the plane of the 

 bend. An attempt to indicate this variation by shading has been made 



* In order that the relative permittivity of the medium be nowhere less than 

 unitj^ the constant «o in the expression e = eo(l + 5) must be greater than the 

 permittivity of free space; but this does not affect the analysis in any wa}^ 



